The energy-moisture balance climate model we
employ is based upon the vertically integrated energy-moisture
balance equations, in which advection terms are replaced
by an eddy-diffusive approximation,
over the oceans, the energy balance
(in ice-free and ice covered regions),
is expressed by,
over land we assume no heat or moisture storage, and hence we write
where is a constant surface air density, the specific heat capacity of air, a constant scale height depth representative of the atmosphere, and the surface air temperature. The terms on the right hand side of (2.1.1) are the various sources/sinks of heat into/out of the system. Here
is the eddy-diffusive horizontal heat transport parameterization, in which is the longitude, is the latitude, a is the radius of the earth, and is a latitudinally dependent heat transport coefficient.
In general, the atmosphere may absorb upwards of 30 % of the
total incoming shortwave radiation through the combined effects
of water vapour, dust, ozone and clouds
(Ramanathan, 1987). To mimic these effects, we apply
a source term in the atmosphere:
where is the solar constant, S is the annual distribution of heat flux entering the top of the atmosphere (North, 1975), is the albedo, and is a reduction parameter representing the scattering/absorption processes described above. Over land, all shortwave radiation intercepted is assumed returned (via black body radiation) to the atmosphere so that assumes a value of zero there.
The net longwave relaxation to space is
modeled by considering the planet as a grey body
. The infrared
emission is then given by
where is the Stefan-Boltzmann constant. Alternatively the moisture longwave feedback effect can be taken into account by employing the planetary longwave parameterization of Thompson and Warren (1982).
where , r is the relative humidity, and are empirically derived constants (see Thompson and Warren, 1982, Table 3).
The longwave radiation
emitted by the ocean is strongly absorbed by greenhouse
gases present in the atmosphere. The
atmosphere then re-emits this absorbed energy both
upward and downward resulting in a longwave
flux at the base of the atmosphere,
which we model as a grey body emission.
The radiative flux is written as
where is the sea, and ice surface temperature, respectively; and are the oceanic, ice, and atmospheric emissivities.
A traditional bulk parameterization is utilized
for the sensible heat flux:
where is the Stanton number, and U is the surface scalar wind speed. The latent heat flux into the atmosphere takes the form
where is the latent heat of evaporation, syr is the number of seconds in a year, and P is the precipitation (in meters per year - assumed to occur when saturation exceeds 85 %).
A moisture balance equation has also been
added so that the atmospheric hydrological cycle
can be included. We obtain a parameterization of
cycle by considering an approximation to the
balance equation for water vapor in the atmosphere.
In this approximation
we replace the horizontal advection terms by an
eddy diffusive term. Vertically integrating over
the depth of the
atmosphere we obtain
where is a constant scale height depth for the specific humidity, , is an eddy diffusive horizontal redistribution term, P is the precipitation, and E is the evaporation (ablation over ice is neglected in the moisture source terms, but retained in the ice heat budget). The evaporation is calculated from its traditional bulk formula
where is the Dalton number, is the saturation specific humidity at (calculated by appealing to the Clausius-Clapeyron equation and utilizing the empirical formula of Bolton, 1980):
Traditionally, a constant humidity is used in the bulk parameterization for the evaporation (e.g. Haney, 1971), however, since we also consider the conservation of water vapor, this is not necessary in the present formulation.
To obtain closure of (2.1.8) we parameterize
the precipitation as
where is the model time step, is the saturation specific humidity at , r is the relative humidity, and